A Proposal to the Seed Money Fund
Cardiopulmonary Resuscitation Using Optimal Control
E. Jung,1 S. M. Lenhart,1,2 V. A. Protopopescu,1,2 and C. F. Babbs3
1Computer Science and Mathematics Division
2Center for Engineering Science Advanced Research
3Basic Medical Sciences, Purdue University
Requested Budgets and Duration
FY 2002 Budget: $125,000
Total Budget: $125,000
Abstract
We propose to develop a new Cardiopulmonary Resuscitation (CPR) strategy to significantly improve artificially maintained circulation of blood with respect to current CPR practice. The new strategy will be realized by applying optimal control (OC) theory to a validated circulation model. The OC approach will maximize the blood flow through vital organs by suitably adjusting the compression rate and/or duration in the validated model. The proposed research will provide a proof-of-principle for the enhanced efficiency of the controlled CPR technique over the traditional ad-hoc techniques used today. This result will be used and extended in more refined circulation models to be addressed in the follow-up funding phases.
Background
Each year, more than 250,000 people die in the United States alone from cardiac arrest.1 Physicians and rescue personnel have long used CPR as the method of choice for maintaining a reasonable fraction of normal blood flow through vital organs during cardiac arrest. Despite widespread use of CPR, the long-term survival for patients with cardiac arrest remains dismal. Indeed, for patients who arrest in-hospital, the rate of long-term survival is 10-15%.28 For out-of-hospital arrest, where CPR is the only procedure immediately available, the odds are much grimmer, namely around 3%.7 The rationale underlying this proposal is that modified methods of resuscitation that generate greater blood flow during CPR can improve long term survival from cardiac arrest - a proposition for which there is already clinical evidence.27
Although there has been considerable interest in studying the mechanism of blood flow during CPR, the practiced techniques have changed relatively little since the 1960’s. Most existing computer simulations of CPR use an electrical lumped parameter model of the circulation,3,4,5,10 governed by a system of ordinary differential equations (ODEs). Various mathematical models3 describe the standard CPR technique and various alternative CPR techniques such as: (i) interposed abdominal compression (IAC), (ii) active compression-decompression, and (iii) Lifestick CPR. Since all these models use time independent parameters (for instance, a fixed compression rate 80 bpm is chosen during the entire CPR cycle), they did not and could not account for time dependent control strategies, even though the condition of the patient deteriorates rapidly with successive minutes of unsuccessful resuscitation.2 Hence in real arrests, the patient-dependent factors such as vascular resistance and compliance are likely to change with time. Moreover, drugs such as epinephrine may be given, which can have dramatic effects upon the circulatory system, lasting for several minutes. Based on our extensive previous work on OC of various lumped and/or distributed systems, we are convinced that this situation can be significantly improved, by bringing the suitable tools to the table.
Objective
We propose to apply OC techniques on a validated circulation model to develop new effective methods for improving hemodynamic efficacy of CPR techniques. The rescuer-dependent factors such as compression rate or compression duration, that significantly affect the magnitude of blood flow during CPR, will be considered time dependent control functions. These functions will be determined to maximize the blood flow through vital organs. In turn, this will result in more effective and successful CPR techniques.
The goals and expected results of this seed money project are: (i) to determine whether OC approaches can be used to significantly increase the blood flow in the circulation system; (ii) to use the results to apply for an National Institutes of Health (NIH) grant; and (iii) to integrate our research within related programs, (e.g., the Digital Human, previously known as the Virtual Human initiative).
Proposed Approach
We shall use an existing validated multicompartment lumped parameter model to calculate pressures and flows during CPR. The chosen model has been developed and validated against real data by Charles F. Babbs, M.D., Ph.D.,3 who is a collaborator on this project. The advantages of this circulation model over similar models are: (i) it has lowest dimensionality; and (ii) provides excellent comparison with real data.
Babbs’ model is a lumped parameter model for the circulatory system, wherein the heart and blood vessels in various parts of the body are represented as resistance-capacitive networks, similar to electric circuits. Following the analogy with Ohm’s law, pressures in the chest, abdomen, and vascular compartments are interpreted as voltages, blood flow as an electric current, and cardiac and venous valves as diodes - electrical devices that permit current flow in only one direction. The analog of the capacitance is the compliance C, defined as where is the incremental change in pressure within a compartment as volume is introduced. Figure 1 shows an example of the elements of the electrical model. Three major sections, namely head, thorax, and abdomen, are considered.
In the state system, the temporal variation of the pressure is calculated for each compartment from a system of ODEs. These equations are derived from the fundamental properties of the circulatory system, namely the relation between pressure gradient and flow (Ohm’s law) and the definition of compliance. The state system based on Babbs’s model3 may be represented as follows:
Extra-thoracic Components:
Thoracic Components:
The valves, indicated by green arrows in figure 1, lead to discontinuous functions depending on the values of ’s in our system. In the circulation system above, the step function q() represents a diodic valve, that is, q() = 1 if and = 0 otherwise. The function q() describes the action of venous valves (Niemann’s valves) at the thoracic inlet. The functions q() and q() represent the actions of the aortic and pulmonic valves, respectively.
The terms and represent driving intrathoracic and intraabdominal pressures applied to outer surfaces of blood vessels in the abdomen and chest by virtue of external compression of either the chest or the abdomen of the victim by the rescuer. Although any arbitrary function or waveform can be used for these external forces, for illustration we shall refer to periodic (sinusoidal) functions. A full compression/relaxation cycle, of length T, comprises a compression period, of length a < T, and a relaxation period, of length T - a. The compression rate is given by. In the standard CPR, we take > 0 during compression and (t) = 0 during relaxation; we note that (t) = 0 during the entire cycle. In IAC-CPR, the chest compression is the same as in the standard CPR. The abdomen compression starts with a relaxation period, followed by a compression waveform (t) = -b sin (wt /2a), which is positive during abdominal compression and zero otherwise. The parameters a and b are the maximum amplitude of the chest and abdomen compression, respectively. The other parameters are summarized in Table 1.
Pressures, Compliances / ResistancesAbdominal aorta / , / Aorta /
Inferior vena cava / / Subphrenic organs /
Carotid artery / / Subphrenic vena cava /
Jugular veins / / Carotid arteries /
Thoracic aorta / / Head + arm resistance /
Right heart &Superior vena cava / / Jugular veins /
Chest pump / / Pump input (tricuspid valve) /
Pump output (aortic valve) /
Coronary vessels /
Table 1. Model Parameters
The main goal of CPR is to restart and maintain a reasonably high blood flow in the vital organs until the heart resumes its normal functions or definite therapy, such as electrical ventricular defibrillation, is applied. In fact, one would like to maximize the blood flow, by adjusting the compression parameters. This formulation is very congenial to the formulation of OC problems. The OC of ODEs was essentially developed by Pontryagin25 and has since been successfully extended and applied to many physical, engineering, ecological, and biological situations. Time dependent control strategies have been studied for HIV immunology models9,15 and a two-strain tuberculosis epidemic model. 13
We shall choose the (time dependent) controls among the crucial rescuer-dependent factors, namely the shape of the compression function and the compression duration. The objective functional, J, to be maximized, is an integral over time of the total blood flow, expressed as a function of the state P, the chosen controls, and - possibly - of time:
.
The necessary conditions that the optimal solution must satisfy are derived from Pontryagin's Maximum Principle.25 Upon applying this principle, the control problem is converted into the problem of maximizing a Hamiltonian function, H, pointwise, with respect of the chosen control functions. The Hamiltonian is canonically constructed as:
where f is the integrand in J, i = 1,…., n, are the right hand sides in the state system (1) - (2), , i = 1,…., n, are the solutions of the adjoint system that is computed by with at the final time, , and n is the dimensionality of the state system (in our case n = 7). The optimal solution is obtained by solving the optimality system, which is a 2n-dimensional ODE system consisting of the n state equations and the n adjoint equations, coupled together by the (explicit) characterization of the OC, as obtained from the optimality condition . Due to the initial condition in the state equation and the final condition in the adjoint equation, the optimality system has opposite time orientations, which usually present computational challenges.
Research Tasks & Deliverables
Task 1. Control Synthesis – New Aspects due to Discontinuities: Some elements of the state system have discontinuities (depending on the states) caused by the motions of the cardiac and venous valves. Since the standard theory covers only smooth functions, the OC theory will be extended to this situation. The OC will be determined for various strategies and several choices of features to be controlled.
Task 2. Computational Modeling: We will implement numerically the optimal strategies for standard or one alternative CPR technique. Various important rescuer-dependent factors such as frequency and compression duration of the external driving intrathoracic or intraabdomenal forces, will be chosen as the time dependent control functions. Due to the opposite time orientations of the equations involved, an alternate iterative method will be used for solving the resulting optimality systems.
Task 3. Validation and Analysis of Results: Babbs’s CPR model3 has been calibrated to an actual physical electrical circuit and validated in dog studies20,26 and human clinical trials.23 The proposed research provides the optimal CPR strategy using the time dependent control functions based on this validated model. After numerically solving the optimality systems, pressure and flow curves in the different compartments will be compared for the blood circulation during CPR with and without controls, to ascertain the improvement obtained by using OC.
Risks vs. Benefits
Our proposal starts from a simple idea: The very low success rate of existing CPR techniques makes us believe that there is ample room for improvement. Clinical studies of alternative CPR methods show that, under certain conditions, at least twice standard survival rates can be obtained with nonstandard methods that have been shown in the laboratory to improve blood flow during cardiac arrest.24,27 These methods however, are ad-hoc and certainly not optimal. Systematic comparisons of various CPR methods in the SAME experimental model are quite rare.3 Hence it is reasonable that the development of a sophisticated mathematical test bed can be of value in suggesting better resuscitation methods for the twenty first century. On the other hand, it may turn out that the OC applied to the circulation model discussed above would not significantly improve the blood flow.
In our opinion, this risk is strongly outweighed by the benefits of a positive outcome. First, our own experience shows that OC problems solved numerically with iterative methods and applied to a host of physical, engineering, biological, and economic situations does indeed result in substantial improvement over ad-hoc procedures. Second, for the present application, even a minor (10-15 %) increase in blood flow may represent the difference between life and death for many of the 250,000 people who presently die prematurely of cardiac arrest. We expect that the controlled CPR model developed in this seed money project will rapidly become an essential tool for understanding complex circulatory physiology during CPR and developing effective CPR techniques, and will have a significant impact on computational sciences and biomedical engineering sciences.
Follow-On Funding
Most biomedical problems today are best addressed using a multi-disciplinary approach that extends beyond traditional physiology and clinical sciences. Multi-disciplinary research based on principles and techniques drawn from mathematics, physics, computer sciences, and engineering is providing deeper understanding and original results that improve human health and quality of life. Recognizing the importance of research in public health, future opportunities in biomedical area are imminent in the Bioengineering Consortium (BECON) at the NIH. The proposed research is well aligned to BECON’s mission. Follow-up funding on the simulations of blood flow during CPR has been discussed with Richard E. Swaja, who is a program manager of BECON at the NIH. The follow-up proposal will be submitted to either the National Institute of General Medical Sciences (NIGMS) or the National Institute of Nursing Research (NINR). We have been in contact with James Cassatt, who is the director of the division of Cell Biology and Biophysics at NIGMS and Hilary Sigmon, who is a program manager of the Cardiopulmonary Health and Critical Care Division at NINR. The follow-upNIH proposal will extend this proof of principle to a more refined circulation model, based on PDEs & ODEs and capable of accounting for additional specific features of the patient.
We realize of course, that while actually performing clinical CPR today there is no direct way to measure blood flow in a routine manner. In fact, carrying out such direct measurements is a major technical objective of the Post-Resuscitation and Initial Utility in Life Saving Efforts initiative and of non-governmental institutions (e.g., Institute for Critical Care Medicine in Palm Springs, California) that will be developed independently and in parallel with our study. However, indirect measures of blood flow, such as carbon dioxide excretion (“end tidal ”), oxygen blood content by clip-on ear sensors, or pressure measurements carried out in the hospital, under monitored circumstances, can be used as approximate measures of the blood flow. Despite the present lack of routine clinical measures of blood flow in cardiac arrest and CPR (except a few ongoing research protocols), we think that it is reasonable to explore, develop, and validate control strategies against existing data. Data from direct measurements on CPR patients themselves would be used as soon as they become available and the two technologies would be merged on a routine basis.